Vol 7, No 5 (2016)

Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 7(5), May – 2016 39

CHEMICAL REACTION EFFECT ON UNSTEADY MHD FLOW THROUGH POROUS MEDIUM

PAST AN EXPONENTIALLY ACCELERATED INCLINED PLATE WITH VARIABLE

TEMPERATURE AND MASS DIFFUSION IN THE PRESENCE OF HALL CURRENT

U. S. RAJPUT, GAURAV KUMAR*

Department of Mathematics and Astronomy,

University of Lucknow - (226007), Lucknow, (U.P.), India.

(Received On: 15-04-16; Revised & Accepted On: 08-05-16)

ABSTRACT

C

hemical reaction effect on unsteady MHD flow through porous medium past an exponentially accelerated inclined plate with variable temperature and mass diffusion in the presence of Hall current is studied here The fluid taken is electrically conducting. The Governing equations involved in the present analysis are solved by the Laplace-transform technique. The velocity profile is discussed with the help of graphs drawn for different parameters like thermal Grashof number, mass Grashof Number, Prandtl number, chemical parameter, Hall current parameter, permeability parameter, phase angle, the magnetic field parameter and Schmidt number; and the numerical values of skin-friction and sherwood number have been tabulated.

Keywords: MHD flow exponentially accelerated plate, variable temperature, mass diffusion, Hall current.

INTRODUCTION

© 2016, IJMA. All Rights Reserved 40 MATHEMATICAL ANALYSIS

MHD flow between two parallel electrically non conducting plates inclined at an angle α _{from vertical is considered. x} axis is taken along the plate and z normal to it. A transverse magnetic field B0 of uniform strength is applied on the

flow. Initially it has been considered that the plate as well as the fluid is at the same temperature T_∞. The species concentration in the fluid is taken as C∞. At time t > 0, the plate starts exponentially accelerating in its own plane with velocity u=u0.ebt, and temperature of the plate is raised to Tw. The concentration C near the plate is raised linearly with

respect to time. The flow modal is as under:

,

)

1

(

)

(

)

(

)

(

₂ 2 0 * 2 2

K

u

m

mv

u

B

C

C

Cos

g

T

T

Cos

g

z

u

t

u

υ

ρ

σ

α

β

α

β

υ

−

+

+

−

−

+

−

+

∂

∂

=

∂

∂

∞

∞ (1)

,

)

1

(

)

(

2 2 0 2 2

K

v

m

v

mu

B

z

v

t

v

υ

ρ

σ

υ

−

+

−

+

∂

∂

=

∂

∂

(2)

),

(

2 2 ∞

−

−

∂

∂

=

∂

∂

C

C

K

z

C

D

t

C

c (3)

.

2 2

z

T

k

t

T

C

_p

∂

∂

=

∂

∂

ρ

(4)

The corresponding initial and boundary conditions are

,

,

,

0

,

0

:

0

=

=

=

_∞

=

_∞

≤

u

v

T

T

C

C

t

for all z

,

)

(

,

)

(

,

0

,

:

0

2 0 2 0

0

υ

υ

t

u

C

C

C

C

t

u

T

T

T

T

v

e

u

u

t

>

=

bt

=

=

_∞

+

_w

−

_∞

=

_∞

+

_w

−

_∞

at z=0 (5)

∞

∞

→

→

→

→

v

T

T

C

C

u

0

,

0

,

,

as

z

→

0

.

Here u is the primary velocity, v -the secondary velocity, g-the acceleration due to gravity,

β

-volumetric coefficient of thermal expansion, b-acceleration parameter, t-time, m(=ω_eτ_e) is the Hall parameter with

e

ω -cyclotron frequency of

electrons and

τ

_e-electron collision time, K-the permeability parameter, T-temperature of the fluid,

β

*-volumetric coefficient of concentration expansion, C-species concentration in the fluid,

ν

-the kinematic viscosity,

ρ

-the density,

p

C

-the specific heat at constant pressure, k-thermal conductivity of the fluid, D-the mass diffusion coefficient,

T

_w

-temperature of the plate at z= 0,

C

_w-species concentration at the plate z= 0,

B

₀-the uniform magnetic field,

K

_c -chemical reaction,

σ

- electrical conductivity.

The following non-dimensional quantities are introduced to transform equations (1), (2), (3) and (4) into dimensionless form: , 2 0 2 0 2 0 3 0 * 2 0 2 0 3 0 0 0 0

,

,

,

)

(

)

(

,

)

(

,

,

)

(

,

,

,

,

)

(

)

(

,

,

,

u

b

b

K

u

K

tu

t

C

C

C

C

C

u

C

C

g

G

u

B

M

u

T

T

g

G

k

c

P

D

S

T

T

T

T

u

v

v

u

u

u

zu

z

w w m w r p r c w

υ

υ

υ

υ

β

ρ

υ

σ

βυ

µ

ρυ

µ

υ

θ

υ

=

=

=

−

−

=

−

=

=

−

=

=

=

=

−

−

=

=

=

=

∞ ∞ ∞ ∞ ∞ ∞ (6)

where

u

is the dimensionless primary velocity,

v

-the secondary velocity,

b

-dimensionless acceleration parameter,

t

-dimensionless time,

θ

-the dimensionless temperature,

C

- the dimensionless concentration,

K

- the dimensionless

permeability parameter,

G

_r-thermal Grashof number,

G

_m-mass Grashof number,

K

₀ -chemical reaction parameter,

µ

-the coefficient of viscosity,

P

_r-the Prandtl number,

S

_c-the Schmidt number, M-the magnetic parameter.

Thus the model becomes

,

1

)

1

(

)

(

2 2 2

u

K

m

v

m

u

M

C

Cos

G

Cos

G

z

u

t

u

m

r

₊

−

+

−

+

+

∂

∂

=

∂

∂

_αθ

_α

© 2016, IJMA. All Rights Reserved 41

,

1

0 2 2

C

K

z

C

S

t

C

c

−

∂

∂

=

∂

∂

(9)

.

1

2 2

z

P

t

_r

∂

∂

=

∂

∂

θ

θ

(10)

The boundary conditions become

,

0

,

0

,

0

,

0

:

0

=

=

=

=

≤

u

v

C

t

θ

for all

z

,

,

,

,

0

,

:

0

u

e

v

t

C

t

t

>

=

bt

=

θ

=

=

at

z

=0, (11)

0

,

0

,

0

,

0

→

→

→

→

v

C

u

θ

as

z

→

∞

_.

Dropping bars in the above equations, we get

,

1

)

1

(

)

(

2 2 2

u

K

m

mv

u

M

C

Cos

G

Cos

G

z

u

t

u

m

r

₊

−

+

−

+

+

∂

∂

=

∂

∂

_αθ

_α

(12)

,

1

)

1

(

)

(

2 2 2

v

K

m

v

mu

M

z

v

t

v

₋

+

−

+

∂

∂

=

∂

∂

(13)

,

1

0 2 2

C

K

z

C

S

t

C

c

−

∂

∂

=

∂

∂

(14)

.

1

2 2

z

P

t

_r

∂

∂

=

∂

∂

θ

θ

(15)

The boundary conditions will become

,

0

,

0

,

0

,

0

:

0

=

=

=

=

≤

u

v

C

t

θ

for all z,

,

,

,

0

,

:

0

u

e

v

t

C

t

t

>

=

bt

=

θ

=

=

at z=0, (16)

,

0

,

0

,

0

,

0

→

→

→

→

v

C

u

θ

as

z

→

∞

.

Combining equations (15) and (16), the model becomes

,

2 2

qa

C

Cos

G

Cos

G

z

q

t

q

m

r

+

−

+

∂

∂

=

∂

∂

_αθ

_α

(17)

,

1

0 2 2

C

K

z

C

S

t

C

c

−

∂

∂

=

∂

∂

(18)

.

1

2 2

z

P

t

_r

∂

∂

=

∂

∂

θ

θ

(19)

The boundary conditions are transformed as

,

0

,

0

,

0

:

0

=

=

=

≤

q

C

t

θ

for all z,

,

,

,

:

0

q

e

t

C

t

t

>

=

bt

θ

=

=

at z=0, (20)

,

0

,

0

,

0

→

→

→

C

q

θ

as

z

→

∞

_.

Here q= u + i v,

.

1

1

)

1

(

2

K

m

im

M

a

+

+

−

=

© 2016, IJMA. All Rights Reserved 42 The solution obtained is as under:

,

]

2

[

)

2

1

(

4 2 2

















−

+

=

− t _r

z r r

r

_e

_P

t

P

z

t

P

erfc

t

P

z

t

π

θ

















+ + + + − − −

= ]( 2 ₀)

2 0 2 [ 0 2 ) 0 2 ]( 2 0 2 [ 0 4 0 K t c S z t K t c S z erfc K c S z e K t c S z t K t c S z erfc K K c S z e C

[

]

)]

0

1

(

8

0

)

1

(

10

13

)

0

(

0

2

9

0

[

2

)

0

(

4

)]

2

2

2

2

(

12

)

7

6

(

14

4

2

2

[

2

2

]

0

3

)

0

1

(

2

2

)

1

(

5

13

2

11

[

2

)

0

(

4

)

1

(

4

14

2

2

2

11

2

4

15

2

1

c

S

tK

c

S

at

A

c

S

K

e

c

S

A

A

az

K

c

S

K

c

S

A

c

S

K

e

c

S

K

a

Cos

m

G

r

P

at

r

P

az

A

r

P

A

A

A

r

tP

t

r

P

z

zae

a

Cos

r

G

a

c

S

K

A

z

a

ze

tK

c

S

A

z

a

e

c

S

A

A

zA

c

S

K

a

Cos

m

G

r

P

A

A

r

P

A

z

a

e

zA

a

Cos

r

G

q

e

bt a bz

A

+

−

−

−

+

−

+

−

−

−

+

+

+

−

+

+

+

−

+

−

−

−

−

+

−

+

−

+

−

+

−

+

+

=

− +

π

π

π

π

α

π

π

π

α

α

α

The expressions for the constants involved in the above equations are given in the appendix.

SKIN FRICTION

The dimensionless skin friction at the plate z=0:

y i x z dz dq τ τ + = =       0 ,

Separating real and imaginary part in

0 =       z dz

dq _{, the dimensionless skin – friction components}

0 = =       z dz du x τ and 0 = =       z dz dv y

τ can be computed.

SHERWOOD NUMBER

The dimensionless Sherwood number at the plate z=0 is given by

,

)

4

1

](

[

)

2

4

1

](

[

0 0 0 0 0 0 0 0 0 0

K

K

tS

e

K

t

K

tK

erfc

S

K

S

t

S

K

tK

erfc

z

C

S

c tK c c c z h

_π

− =

−

+

+

−

−

−

=













∂

∂

=

RESULT AND DISCUSSIONS

The velocity profile for different parameters like, thermal Grashof number (Gr), magnetic field parameter (M), Hall parameter (m), Prandtl number (Pr), chemical reaction parameter(K0), acceleration parameter (b) and time (t) is shown

in figures 1.1 to 2.10. The concentration profile for different parameters like, chemical reaction parameter, Schmidt number and time are shown in figure 3.1 to 3.3. It is observed from figures 1.1 and 2.1 that the primary and secondary velocities of fluid decrease when the angle of inclination (α) is increased. It is observed from figure 1.2 and 2.2, when the mass Grashof number Gr is increased then the velocities are increased. From figures 1.3 and 2.3 it is deduced that velocities increases with thermal Grashof number Gr. If Hall current parameter m is increased then u increases, while v gets decreased (figures 1.4 and 2.4). Also, it is observed from figures 1.5 and 2.5 that the effect of increasing values of the parameter M results in decreasing u and increasing v. It is deduced that when chemical reaction parameter K0 is

© 2016, IJMA. All Rights Reserved 43 Skin friction is given in table1. The value of τx increases with, thermal Grashof Number, the mass Grashof number, Hall current parameter, Schmidt number and time, and it decreases with the increase in the angle of inclination of plate, chemical reaction parameter, acceleration parameter, the magnetic field parameter, permeability parameter and Prandtl number. However, similar effect is observed with τy, except in the case of chemical reaction parameter, Hall parameter, the magnetic field parameter and acceleration parameter, in which case effect is reversed.

Sherwood number is given in table 2. The value of Sh decreases with chemical reaction parameter, Schmidt number and

time

Figure-1.1: Velocity u for different values of

α

Figure-1.2: Velocity u for different values of Gm

Figure-1.3: Velocity u for different values of Gr Figure-1.4: Velocity u for different values of m

Figure-1.5: Velocity u for different values of M Figure-1.6: Velocity u for different values of K₀

© 2016, IJMA. All Rights Reserved 44

Figure-1.9: Velocity u for different values of Pr Figure-1.10: Velocity u for different values of Sc

Figure-1.11: Velocity u for different values of t Figure-2.1: Velocity v for different values of

α

Figure-2.2: Velocity v for different values of Gm Figure-2.3: Velocity v for different values of Gr

© 2016, IJMA. All Rights Reserved 45

Figure-2.6: Velocity v for different values of K₀ Figure-2.7: Velocity v different values of b

Figure-2.8: Velocity v different values of K Figure-2.9: Velocity v different values of Pr

Figure-2.10: Velocity v for different values of Sc Figure-2.11: Velocity v for different values of t

© 2016, IJMA. All Rights Reserved 46 Figure-3.3: Concentration c for different values of t

Table-1: Skin friction for different Parameters. α

(in degree) M m Pr Sc Gm Gr K0 K t b

τ

x

τ

y

15 2 1 0.71 2.01 100 10 1 0.2 0.2 1 8.52178 3.23741

30 2 1 0.71 2.01 100 10 1 0.2 0.2 1 7.29792 2.92419

60 2 1 0.71 2.01 100 10 1 0.2 0.2 1 2.81379 1.77658

30 3 1 0.71 2.01 100 10 1 0.2 0.2 1 5.57206 3.19740

30 5 1 0.71 2.01 100 10 1 0.2 0.2 1 3.30955 3.19760

30 2 3 0.71 2.01 100 10 1 0.2 0.2 1 10.7310 3.04336

30 2 5 0.71 2.01 100 10 1 0.2 0.2 1 11.7683 2.20051

30 2 1 7.00 2.01 100 10 1 0.2 0.2 1 7.16722 2.92064

30 2 1 0.71 3.00 100 10 1 0.2 0.2 1 14.7514 8.27996

30 2 1 0.71 4.00 100 10 1 0.2 0.2 1 28.6901 28.3203

30 2 1 0.71 2.01 10 10 1 0.2 0.2 1 -1.99761 0.48486

30 2 1 0.71 2.01 50 10 1 0.2 0.2 1 2.03374 1.56901

30 2 1 0.71 2.01 100 50 1 0.2 0.2 1 8.42268 2.94390

30 2 1 0.71 2.01 100 100 1 0.2 0.2 1 9.82864 2.96853

30 2 1 0.71 2.01 100 10 5 0.2 0.2 1 13.7523 -11.2023

30 2 1 0.71 2.01 100 10 10 0.2 0.2 1 0.98453 -0.47930

30 2 1 0.71 2.01 100 10 10 0.5 0.2 1 13.2100 33.9004

30 2 1 0.71 2.01 100 10 10 1.0 0.2 1 -39.6732 28.9697

30 2 1 0.71 2.01 100 10 1 0.2 0.3 1 12.9159 4.34719

30 2 1 0.71 2.01 100 10 1 0.2 0.4 1 18.6483 5.78354

30 2 1 0.71 2.01 100 10 1 0.2 0.2 3 5.07323 3.00124

30 2 1 0.71 2.01 100 10 1 0.2 0.2 5 1.53083 3.11014

Table-2: Sherwood number for different Parameters.

0

K

Sc t

S

_h

1 2.01 0.2 -0.762200 5 2.01 0.2 -0.933049 10 2.01 0.2 -1.118240 1 3.00 0.2 -0.931175 1 4.00 0.2 -1.075230 1 2.01 0.3 -0.961323 1 2.01 0.4 -1.141570

CONCLUSION

The conclusions of the study are as follows:

• Primary velocity increases with the increase in thermal Grashof number, mass Grashof number, Hall current parameter, permeability parameter, acceleration parameter and time.

© 2016, IJMA. All Rights Reserved 47

• Secondary velocity increases with the increase in thermal Grashof number, mass Grashof number, the magnetic field, permeability parameter, acceleration parameter and time.

• Secondary velocity decreases with the angle of inclination of plate, Hall current parameter, chemical reaction parameter, Prandtl number and Schmidt number.

• τx increases with the increase in Gm, Gr, m, Sc and t, and it decreases withα, b, K0, M, K and Pr.

• τy increases with the increase in Gm, Gr, K0 , M, b, Sc and t, and it decreases withα, K,m and Pr

• Sh decreases with K0, Sc and t.

APPENDIX

),

1

(

1

₁₆ 2 ₁₇

1

A

e

A

A

=

+

+

az

−

A

₂

=

−

A

₁

,

A

₃

=

A

₁₆

−

A

₁

,

A

₄

=

−

1

+

A

₂₂

+

A

₁₈

(

A

₂₃

−

1

),

),

1

(

1

_{24 19 25}

5

=

−

+

A

+

A

A

−

A

A

₆

=

−

1

−

A

₂₆

+

A

₁₈

(

A

₂₇

−

1

),

A

₇

=

−

A

₆

,

A

₈

=

−

1

−

A

₂₀

+

A

₃₀

(

A

₂₁

−

1

),

),

1

(

2

₂₀

8

9

=

A

+

A

+

A

A

₁₀

=

−

1

−

A

₂₈

+

A

₁₉

(

A

₂₉

−

1

),

11

(

2

A

1

2

atA

2

a

A

3

),

z

e

A

az

+

+

=

−

,

2

1

12

















+

−

=

t

P

z

erf

A

r 1 1

_,

) ( 1 13 0 0 c c c c c S S tK S S K a z S at

e

A

−+

− + − − − + −

=

1

,

) ( 1 14 r r r P P a z P at

e

A

−+

− + −

=

A

₁₅

=

1

+

A

₃₁

+

e

2 a+bz

A

₃₂

,

]

2

2

[

16

t

z

t

a

erf

A

=

−

,

],

2

2

[

17

t

z

t

a

erf

A

=

+

r

r

P aP z

e

A

−+

−

=

2 1

18 , c c S S K a z

e

A

−+

− −

=

1 ) ( 2 19 0 ,

_,

2

0 20

















−

=

t

S

z

tK

erf

A

c

,

2

0 21

















+

=

t

S

z

tK

erf

A

c _],

2 1 2 [ 22 t P aP t z erf A r r + − − = ], 2 1 2 [ 23 t P aP t z erf A r r + − + =

],

2

1

)

(

2

[

0 24

t

S

S

K

a

t

z

erf

A

c c

+

−

−

−

=

],

2

1

)

(

2

[

0 25

t

S

S

K

a

t

z

erf

A

c c

+

−

−

+

=

], 2 1 2 [ 26 t P z P a t erf A r r − + − = ], 2 1 2 [ 27 t P z P a t erf A r r + + − =

],

2

1

)

(

[

0 28

t

zS

S

K

a

t

erf

A

c c

−

+

−

−

=

],

2

1

)

(

[

0 29

t

zS

S

K

a

t

erf

A

c c

+

+

−

−

=

],

[

2 0

30 c S K z

e

erf

A

=

],

2

2

[

31

t

z

t

b

a

erf

A

=

+

−

],

2

2

[

32

t

z

t

b

a

erfc

A

=

+

+

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